Introduction to the thermodynamics of small systems and the spin variable method 

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Single-molecule force spectroscopy

As described above, the functions of several macromolecules (for instance, proteins) are strongly related to the three dimensional conformation of their polymeric chain. Studying the relation between the three dimensional conformation of a macromolecule and its function can be directly done by using force spectroscopy methods (see Fig.1.8), which can be used to unfold the native folded structure of a macromolecule. The controlled unfolding leads to the estimate of the involved forces, the energy landscape and many other dynamic properties of the system under investigation. Recent developments of mechanical experiments on single-molecule allowed to better understand intra- and intermolecular forces, by introducing important information about the thermodynamics and kinetics of several molecular processes. Single-molecule experiments are typically based on optical tweezers, magnetic tweezers, microelectromechanical systems (MEMS) and atomic force microscope (AFM) [31]. With such devices, Figure 1.6 Representation of the molecular genesis of Huntington’s disease. Huntington’s disease is a progressive and invariably fatal neurodegenerative genetic disorder. The gene of this disease, Huntingtin, contains a repeat of CAG codon coding for glutamine (a codon is a sequence of three nucleotides on a messenger ribonucleic acid). If the repeat contains 35 or more repeats, Huntington’s disease develops and it results in the death of brain cells, leading to, inter alia, problems with mental abilities and a general lack of coordination [27].

Conventional and high-speed atomic force microscope

The atomic force microscope is a well-known technique invented in 1985 by Gerd Binnig, Calvin Quate and Christoph Gerber and commercialised for the rst time in 1989 [32]. The AFM is a high-resolution scanning probe microscopy instrument allowing to reach the atomic resolution. In its primary operation mode, also known as « contact mode », AFM allows to visualise the topography of a sample surface by scanning it horizontally with a sharp tip placed at the extremity of a cantilever. As high-resolution imaging tool, it permits to measure the roughness of a sample surface.
The two main components of the AFM are the cantilever, which acts as a exible sensor and a piezoelectric positioner, in order to control the sample position in the nanometric range. An AFM consists of a cantilever, mounted on a cantilever holder, whose position is controlled by a piezoelectric device. A focused laser beam is reected o the surface of the cantilever on a photodetector. The moves of the cantilever can be monitored by following the movement of the laser spot on the photodiode. The incident light is converted into voltage by the photodiode output, which then outputs the voltage dierence when the laser spot moves. The AFM measures the angular deviation of the laser spot allowing to obtain the forces exerted on the exible cantilever. This technique has also been generalised to stretch a macromolecule and therefore to perform single-molecule force spectroscopy (SMFS), as observed in Fig.1.9. In this case, the pulling speeds used in SMFS experiments vary in the range of a few nm/s to about 10 m s􀀀1, and the cantilever stiness is typically between 6 and 100 pN nm􀀀1. Recently, a high-speed AFM (HS-AFM) has been developed to unfold proteins at higher velocities than conventional AFM, allowing to reach pulling speeds of the order of 4000 m/s [3438]. This was made possible by using short cantilevers and miniature piezoelectric actuator (see Fig.1.10). This is a signicant development since it permits to SMFS experiments to be comparable to those probed in molecular dynamics simulations, which oer atomic-level descriptions of the forced unfolding [39, 40].

Applications of the spin variable method

In the Section 2.1, we presented some general results of thermodynamics obtained with the help of the spin variable method. Here, the spin variable approach is described in detail, for specic one-dimensional and three-dimensional systems. The study of these two cases has already been published in Ref. [113]. They are here presented to introduce the spin variable method in the simplest cases, without any of the extensions which will be proposed in the following, like elasticity or again Ising interactions. Moreover, we show an example to prove that the spin variables method is not also useful to obtain average values of positions or forces but also to determine the complete probability density describing systems with bistability.

Table of contents :

1 State of the art and motivations 
1.1 Why nanomechanics of macromolecules?
1.1.1 Structural stability of proteins
1.1.2 Dynamics of macromolecules
1.1.3 Thermodynamics of small systems
1.1.4 Mechanical consequences on health
1.2 Single-molecule force spectroscopy
1.2.1 Conventional and high-speed atomic force microscope
1.2.2 Magnetic tweezers
1.2.3 Optical tweezers
1.2.4 MEMS
1.3 DNA, RNA and models
1.3.1 DNA and RNA
1.3.2 Freely jointed chain model and worm-like chain model
1.4 Proteins
1.5 Structures with bistability
1.6 Motivations and goals
2 Introduction to the thermodynamics of small systems and the spin variable method 
2.1 Thermodynamics of small systems
2.1.1 Introduction
2.1.2 Thermodynamics of chains with conformational transitions
2.2 Applications of the spin variable method
2.2.1 One dimensional system
2.2.1.1 The Gibbs ensemble
2.2.1.2 The Helmholtz ensemble
2.2.2 Bistable freely jointed chain
2.2.2.1 The Gibbs ensemble
2.2.2.2 The Helmholtz ensemble
2.3 Full statistics of conjugated thermodynamic ensembles in chains of twostate units
2.3.1 Congurational partition functions and force-extension relations in the Gibbs and the Helmholtz ensembles
2.3.2 Complete probability densities in the Gibbs and the Helmholtz ensembles
2.3.3 Probability density of the couple (x_N; xN) versus f within the Gibbs ensemble
2.3.4 Probability density of the couple (f_; f) versus xN within the Helmholtz ensemble
2.3.5 Final comparison
3 Extensible two-state freely jointed chain 
3.1 Introduction
3.2 Two-state freely jointed chain with extensible units: the Gibbs ensemble
3.3 Two-state freely jointed chain with extensible units: the Helmholtz ensemble
3.3.1 An integral calculation
3.3.2 The Hermite elements with negative index
3.3.3 The partition function and related results
3.4 Conclusion
4 Two-state freely jointed chain with Ising interactions 
4.1 Introduction
4.2 Example of biological cooperativity
4.3 Two-state freely jointed chain with Ising interactions: the Gibbs ensemble
4.4 Two-state freely jointed chain with Ising interactions: the Helmholtz ensemble
4.5 Explicit expression for the Helmholtz response under weak Ising interaction: jj kBT
4.6 Explicit expression for the Helmholtz response under strong Ising ferromagnetic interaction: kBT
4.7 Explicit expression for the Helmholtz response under strong Ising antiferromagnetic interaction: 􀀀kBT
4.8 The thermodynamic limit
4.9 Ising interactions coupled with extensible units
4.10 Conclusion
5 Two-state heterogeneous chains 
5.1 Introduction
5.2 Examples of unfolding pathway
5.3 Two-state heterogeneous one-dimensional system
5.3.1 The Gibbs ensemble
5.3.2 The Helmholtz ensemble
5.4 Unfolding pathway identiability
5.5 Conclusion
6 Pulling speed dependence of the force-extension response of bistable chains 
6.1 Introduction
6.2 Out-of-equilibrium statistical mechanics through the Langevin approach
6.3 Analytical and numerical results
6.3.1 Device without intrinsic elasticity
6.3.2 Device with intrinsic elasticity
6.4 Theory meets experiments
6.4.1 Modelling the dynamic stretching of lamin
6.4.2 Modelling the dynamic stretching of titin
6.5 Conclusion
Conclusions and perspectives
Bibliography

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